Grade 10

Angles of elevation and depression

Lesson

In real life contexts we use special words to describe particular angles.

Angle of Elevation

An angle of elevation is the angle created when an observer is looking at a object which is above the horizontal. The angle between the horizontal and the observer's line of sight is called angle of elevation.

Angle of Depression

An angle of Depression is the angle created when an observer is looking at an object which is below the horizontal. The angle between the horizontal and the observer's line of sight is called angle of depression.

Using the angle of elevation or depression, we can create right-angled triangles. If we know the height above or below we can find the distance an object is from an observer, or vice-versa.

Example

Jasper and Jasmine were playing in the park and wondered how tall the tallest tree would be. Jasper tried climbing the tree, trying to use his tape measure as he climbed. This took a few hours, and was quite tricky in places. Jasmine remembered some stuff about trigonometry from school and measured the distance from the tree and the angle of elevation. These measurements were much easier to get and didn't involve any climbing!

The distance from the base of the tree to a point of observation was $4.2$4.2 m.

The angle of elevation from this point to the top of the tree is $38$38 degrees.

$\tan38^\circ$`tan`38°	$=$=	$\frac{\text{Height of tree }}{4.2}$Height of tree 4.2 m
$\text{height of tree }$height of tree	$=$=	$4.2\times\tan38^\circ$4.2×`tan`38°
$\text{height of tree }$height of tree	$=$=	$3.28$3.28

So the height of this tree is $3.28$3.28 m.

Now, imagine they had to work out the height to the peak of a mountain, or the height of an airplane in the sky. Trigonometry can help with a lot of these. In fact there are many examples of professions that use trigonometry:

Surveyers measure an angle and a length, then use trigonometry to get another length that cannot be measured directly, such as a distance across a lake.
Astronomers measure the lengths of shadows of mountains on the moon and knowing the sun angle can compute the heights of the mountains. They also use it when finding the distance between celestial bodies
Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles
Ship Captains use trigonometry in navigation to find the distance of the shore from a point in the sea.
Oceanographers use trigonometry when calculating the height of tides in oceans

Let's have a look at these worked examples.

Question 1

A man standing at point $C$C, is looking at the top of a tree at point $A$A. Identify the angle of elevation in the figure given.

A right triangle is shown with vertices labeled A, B and C. Side AB is the vertical leg, side BC is the horizontal leg, and side AC is the hypotenuse. The right angle is at vertex B, as indicated by a small square. There are two arcs indicating angles: one at vertex C, labeled with $\alpha$`α`, and another at vertex A, labeled with $\theta$`θ`. Above vertex C, a vertical dotted line extends from vertex C to a point labeled D. The angle between this dotted line CD and the hypotenuse AC is labeled with $\sigma$`σ`.

$\alpha$α
A
$\theta$θ
B
$\sigma$σ
C

Question 2

Considering the diagram below, find $x$x, the angle of depression from point $B$B to point $C$C.

Round your answer to two decimal places.

Rectangle $ABDC$`ABDC` is shown. $AC$`AC` is the horizontal base measuring $3$3 units. The horizontal top side $BD$`BD`, opposite to $AC$`AC`, is drawn as dotted line. Side $AB$`AB` and $CD$`CD` are the vertical sides of the rectangle and opposite each other. The angles at vertices $A$`A` and $D$`D`, $\angle BAC$∠`BAC` and $\angle BDC$∠`BDC`, are marked with small boxes indicating a right angle. A diagonal $BC$`BC` is drawn and measures $5$5 units, dividing the rectangle into two right-angled triangles, $\triangle BAC$△`BAC` and $\triangle BDC$△`BDC`. The two triangles share a hypotenuse which is the diagonal $BC$`BC`. $\angle DBC$∠`DBC`, the angle of depression from point $B$`B` to point $C$`C`, is labeled as $x$`x` degrees and marked with a double arc. The x-degree angle is adjacent side $BD$`BD`. The x-degree angle is opposite side $CD$`CD`. $\angle ABC$∠`ABC` is labeled as $y$`y` degrees and marked with a single arc.

Question 3

At a certain time of the day a light post, $6$6 m tall, has a shadow of $5.8$5.8 m. If the angle of elevation of the sun at that time is $\theta$θ°, find $\theta$θ to two decimal places.

Question 4

From the top of a rocky ledge $188$188 m high, the angle of depression to a boat is $13^\circ$13°. If the boat is $d$d m from the foot of the cliff find $d$d correct to two decimal places.

Outcomes

10D.T2.03

Solve problems involving the measures of sides and angles in right triangles in real life applications, using the primary trigonometric ratios and the Pythagorean theorem.